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Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics (Popular Science)

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Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics (Popular Science) Cover

 

Synopses & Reviews

Publisher Comments:

This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition of mathematic fundamentals. Rather than a system of theorems with completely developed proofs or examples of applications, readers will encounter a coherent presentation of mathematical ideas that begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of the calculus.

Contents include examinations of the various types of numbers and a criticism of the extension of numbers; arithmetic, geometry, and the rigorous construction of the theory of integers; the rational numbers, the foundation of the arithmetic of natural numbers, and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers.

In issues of mathematical philosophy, the author explores basic theoretical differences that have been a source of debate among the most prominent scholars and on which contemporary mathematicians remain divided. "With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics." — Carl B. Boyer, Brooklyn College. 27 figures. Index.

Book News Annotation:

This survey of mathematical concept formation begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of calculus. No formal training in mathematics is necessary. This is an unabridged republication of the edition published by Harper & Row, New York, 1959. Annotation (c)2003 Book News, Inc., Portland, OR (booknews.com)

Synopsis:

Examinations of arithmetic, geometry, and theory of integers; rational and natural numbers; complete induction; limit and point of accumulation; remarkable curves; complex and hypercomplex numbers; more. Includes 27 figures. 1959 edition.

Synopsis:

Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; and more. Includes 27 figures. Index. 1959 edition.

Synopsis:

Presupposing no formal training, this volume offers a clear exposition of the fundamentals of mathematics. It starts with the natural numbers and basic laws of arithmetic, progressing to the problems of the real-number continuum and concepts of the calculus. Contents: 1. The Various Types of Numbers. 2. Criticism of the Extension of Numbers. 3. Arithmetic and Geometry. 4. The Rigorous Construction of the Theory of Integers. 5. The Rational Numbers. 6. Foundation of the Arithmetic of Natural Numbers. 7. Rigorous Construction of Elementary Arithmetic. 8. The Principle of Complete Induction. 9. Present Status of the Investigation of the Foundations. 10. Limit and Point of Accumulation. 11. Operating with Sequences. Differential Quotient. 12. Remarkable Curves. 13 The Real Numbers. 14 Ultrareal Numbers. 15. Complex and Hypercomplex Numbers. 16. Inventing or Discovering. Index. Unabridged republication of the edition published by Harper & Row, New York, 1959.

Table of Contents

Foreward

Author's Preface

1. The Various Types of Numbers

2. Criticism of the Extension of Numbers

3. Arithmetic and Geometry

4. The Rigorous Construction of the Theory of Integers

5. The Rational Numbers

6. Foundation of the Arithmetic of Natural Numbers

7. Rigorous Construction of Elementary Arithmetic

8. The Principle of Complete Induction

9. Present Status of the Investigation of the Foundations

  A. Formalism

  B. The Logical School

  C. Outlook

10. Limit and Point of Accumulation

11. Operating with Sequences. Differential Quotient

12. Remarkable Curves

  Appendix: What is Geometry?

13. The Real Numbers

  A. Cantor's Theory

  B. Dedekind's Theory

  C. Comparison of the Two Theories

  D. Uniqueness of the Real Number System

  E. Various Remarks

14. Ultrareal Numbers

15. Complex and Hypercomplex Numbers

16. Inventing or Discovering

Epilogue

Index

Product Details

ISBN:
9780486428048
Author:
Waismann, Friedrich
Publisher:
Dover Publications
Location:
Mineola, N.Y.
Subject:
General
Subject:
Philosophy
Subject:
Mathematics
Subject:
Number concept
Subject:
General Mathematics
Subject:
Mathematics -- Philosophy.
Subject:
Logic
Subject:
Mathematics - General
Edition Number:
Dover ed.
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
576
Publication Date:
20030431
Binding:
TRADE PAPER
Language:
English
Illustrations:
27
Pages:
272
Dimensions:
8.5 x 5.38 in 0.67 lb

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Related Subjects


Reference » Science Reference » General
Science and Mathematics » Mathematics » Foundations and Logic
Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Logic and Philosophy

Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics (Popular Science) New Trade Paper
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Product details 272 pages Dover Publications - English 9780486428048 Reviews:
"Synopsis" by ,
Examinations of arithmetic, geometry, and theory of integers; rational and natural numbers; complete induction; limit and point of accumulation; remarkable curves; complex and hypercomplex numbers; more. Includes 27 figures. 1959 edition.

"Synopsis" by ,
Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; and more. Includes 27 figures. Index. 1959 edition.

"Synopsis" by , Presupposing no formal training, this volume offers a clear exposition of the fundamentals of mathematics. It starts with the natural numbers and basic laws of arithmetic, progressing to the problems of the real-number continuum and concepts of the calculus. Contents: 1. The Various Types of Numbers. 2. Criticism of the Extension of Numbers. 3. Arithmetic and Geometry. 4. The Rigorous Construction of the Theory of Integers. 5. The Rational Numbers. 6. Foundation of the Arithmetic of Natural Numbers. 7. Rigorous Construction of Elementary Arithmetic. 8. The Principle of Complete Induction. 9. Present Status of the Investigation of the Foundations. 10. Limit and Point of Accumulation. 11. Operating with Sequences. Differential Quotient. 12. Remarkable Curves. 13 The Real Numbers. 14 Ultrareal Numbers. 15. Complex and Hypercomplex Numbers. 16. Inventing or Discovering. Index. Unabridged republication of the edition published by Harper & Row, New York, 1959.
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